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Proof-theoretic analysis of KPM

Michael Rathjen
1991 Archive for Mathematical Logic  
In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength of KPM.  ...  KPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets.  ...  I also profited from a "work-out" of [2] given by Ludwig Tautz. Thanks are due to Professor Pohlers for stimulating conversations about proof theory.  ... 
doi:10.1007/bf01621475 fatcat:yxcbpnlu3nggpax47suqhvutni

Proof Theoretic Analysis by Iterated Reflection [chapter]

L. D. Beklemishev
2015 Turing's Revolution  
We compare the information obtained by this kind of analysis with the results obtained by the more usual proof-theoretic techniques.  ...  It has to be noted, however, that the standard proof-theoretic methods, in practical cases, usually allow to define natural ordinal notation systems for suitable initial segments of the constructive ordinals  ...  Thus, the calculation of proof-theoretic ordinals, or ordinal analysis, has become one of the central aims in the study of formal systems (cf. [22] ).  ... 
doi:10.1007/978-3-319-22156-4_9 fatcat:ogd5nz5h3vd23ef4fzfjcwqgdi

Proof-theoretic analysis by iterated reflection

L.D. Beklemishev
2003 Archive for Mathematical Logic  
We compare the information obtained by this kind of analysis with the results obtained by the more usual proof-theoretic techniques.  ...  It has to be noted, however, that the standard proof-theoretic methods, in practical cases, usually allow to define natural ordinal notation systems for suitable initial segments of the constructive ordinals  ...  Thus, the calculation of proof-theoretic ordinals, or ordinal analysis, has become one of the central aims in the study of formal systems (cf. [22] ).  ... 
doi:10.1007/s00153-002-0158-7 fatcat:4ivmz7xtgbhntd7khx6ostj23y

Proof Theoretic Complexity [chapter]

G. E. Ostrin, S. S. Wainer
2002 Proof and System-Reliability  
In particular, Leivant's (1995) theories based on his ramified induction schemes give proof theoretic characterizations of PTIME and the low Grzegorczyk classes E 2 and E 3 .  ...  Previous results of Leivant are reworked and extended in this new setting, with quite different proof theoretic methods. computable.  ...  It will be convenient, for later proof theoretic analysis, to work with logic in a sequent-style formalism, and the system G3 (with structural rules absorbed) as set out on page 65 of Troelstra and Schwichtenberg  ... 
doi:10.1007/978-94-010-0413-8_12 fatcat:mo6ha6scmzckrljxoqfuqidsae

A Game-Theoretic Computational Interpretation of Proofs in Classical Analysis [article]

Paulo Oliva, Thomas Powell
2012 arXiv   pre-print
It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the  ...  that can be clearly understood in game theoretic terms.  ...  functionals and therefore exists in any model of system T.  ... 
arXiv:1204.5244v1 fatcat:d5joerlcgveehgdskqc6yfacuq

Proof theory and computational analysis

Ulrich Kohlenbach
1998 Electronical Notes in Theoretical Computer Science  
In this survey paper we start with a discussion how functionals of nite type can be used for the proof-theoretic extraction of numerical data (e.g. e ective uniform bounds and rates of convergence) from  ...  non-constructive proofs in numerical analysis.  ...  So PB can be considered as a (very weak) system of polynomially bounded arithmetic.  ... 
doi:10.1016/s1571-0661(05)80219-1 fatcat:ixrvz2n45jbjffobwhegqgyfxq

Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE [article]

Martin Fischer, Carlo Nicolai, Pablo Dopico Fernandez
2020 arXiv   pre-print
Questions concerning the proof-theoretic strength of classical versus non-classical theories of truth have received some attention recently.  ...  A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics.  ...  then conclude that the systems of ramified analysis below Γ 0 are proof-theoretically reducible to our system KFL * .  ... 
arXiv:2007.07188v3 fatcat:s3r5tmrnhfbklh4xr3jngcdvdu

The proof-theoretic analysis of Σ11 transfinite dependent choice

Christian Rüede
2003 Annals of Pure and Applied Logic  
This article provides an ordinal analysis of 1 1 transÿnite dependent choice.  ...  Finite reduction In this and the next section the proof-theoretic analysis of E n is given.  ...  The theory 1 1 -TDC 0 and its proof-theoretic analysis typically belong to the new area of so-called metapredicative proof theory.  ... 
doi:10.1016/s0168-0072(03)00006-x fatcat:sphmyymtrfbglklqmutfhhvldm

Proof-Theoretic Analysis of The Quantified Argument Calculus

Edi Pavlović, Norbert Gratzl
2019 The Review of Symbolic Logic  
this out. reason to use sequent calculus in this analysis is to provide a constructive proof of consistency, but first and foremost to prove an important, useful and interesting result of a cut elimination  ...  Ben-Yami makes use of natural deduction (Suppes-Lemmonstyle), we, however, have chosen a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results  ...  Given that this is an interesting but not widely known system, we will present it here in considerable detail before proceeding with the proof-theoretic analysis of it.  ... 
doi:10.1017/s1755020318000114 fatcat:ke464pghrrhhdle4mrktac4d6y

Proof-theoretic strengths of weak theories for positive inductive definitions [article]

Toshiyasu Arai
2018 arXiv   pre-print
In this paper the lightface Π^1_1-Comprehension axiom is shown to be proof-theoretically strong even over RCA_0^*, and we calibrate the proof-theoretic ordinals of weak fragments of the theory ID_1 of  ...  Conjunctions of negative and positive formulas in the transfinite induction axiom of ID_1 are shown to be weak, and disjunctions are strong.  ...  Our proof of Theorem 2.4 is based on an analysis through the operator controlled derivations due to W. Buchholz [3] . An ordinal notation system with the ψ-function also due to W.  ... 
arXiv:1603.01342v7 fatcat:sddiwn2iunenlje3fjcsfeqmzi

A proof-theoretic semantic analysis of dynamic epistemic logic

Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano, Vlasta Sikimić
2014 Journal of Logic and Computation  
The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics.  ...  Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic  ...  that will arise from the proof-theoretic analysis.  ... 
doi:10.1093/logcom/exu063 fatcat:im7schyimraarauhbockei3dim

A Proof-Theoretic Analysis of Theories for Stratified Inductive Definitions [chapter]

Gerhard Jäger, Dieter Probst
2015 Gentzen's Centenary  
In this article we study subsystems SID ν of the theory ID 1 in which fixed point induction is restricted to properly stratified formulas.  ...  Before turning to the proof-theoretic analysis of the systems H ν , we need some further auxiliary notions.  ...  P This finishes our proof-theoretic analysis of the theories SID ν .  ... 
doi:10.1007/978-3-319-10103-3_15 fatcat:shpvcwbcc5anjlv27a4bqzcjnq

On Relating Theories: Proof-Theoretical Reduction [chapter]

Michael Rathjen, Michael Toppel
2019 Mathesis Universalis, Computability and Proof  
The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition.  ...  devised for the purpose of formalizing and presenting various chunks of mathematics.  ...  Both, the support and the hospitality of HIM, are gratefully acknowledged.  ... 
doi:10.1007/978-3-030-20447-1_16 fatcat:loptt6g2jfbmtdczu2qypbk2vu

Essence As A Modality: A Proof-Theoretic and Nominalist Analysis

Preston Stovall
2021 Philosophers' Imprint  
This essay adapts recent developments in proof-theoretic semantics to provide a nominalist analysis for a variety of essentialist vocabularies.  ...  On this proposal, object-language essentialist vocabulary is mentioned in a proof-theoretic metalanguage that uses a vocabulary of explanation.  ...  I will mention two classes of question left open by this analysis: The first concerns details of the underlying proof system, and the second concerns the use of that proof system to interpret talk of essence  ... 
doi:10.3998/phimp.3521354.0021.007 fatcat:xlukbv22xncnbdtuikwervpvfi

Proof-theoretic investigations on Kruskal's theorem

Michael Rathjen, Andreas Weiermann
1993 Annals of Pure and Applied Logic  
Rathjen, M. and A. Weiermann, Proof-theoretic investigations on Kruskal's theorem, Annals of Pure and Applied Logic 60 (1993) 49-88.  ...  Weiermann Our paper gives a complete proof-theoretic characterization of Kruskal's theorem in terms of ordinal notation systems, subsystems of second-order arithmetic, and subsystems of Kripke-Platek set  ...  An ordinal notation system Firstly, we need some ordinal-theoretic background. Let On be the class of ordinals.  ... 
doi:10.1016/0168-0072(93)90192-g fatcat:3y5qaamjwvdo7hoqnlovuruy3y
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